1. Choice

2. Economic Rationality
Assumption: The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it (i.e. consumers choose the most preferred bundle from their budget sets).

3. Utility Maximization Problem (with two goods)
First, we will use the graphical demonstration to solve the problem
Second, we will use mathematical methods for Constrained Maximization Problems (discussed in Math-review slides) to solve the problem

4. Notations and Definitions
The most preferred affordable bundle is called the consumer’s DEMAND at the given prices and budget.
Demands (or optimal choice) will be denoted by
(x1*, x2*)
When x1* > 0 and x2* > 0 the demand (or optimum) is an Interior Solution
If either x1* = 0 or x2* = 0, then the demand (or optimum) is a Corner Solution.
If buying (x1*,x2*) costs $m then the budget is exhausted.

5. Utility Maximization Problem and Optimal Choice (with two goods case) – Graphical Method
Let’s try to find the most preferred affordable (optimal or demanded) bundle graphically by assuming that the Indifference curves are smooth (i.e. Indifference curves have convex decreasing shape).
NOTE: As long as one of the goods, at least, is the good that the consumer likes, then the optimal bundle should be on the budget line.

6. Optimal Choice (with two goods case) – Graphical Methodx2
(x1*,x2*) is the most preferred affordable bundle.
(x1*,x2*) is interior
(x1*,x2*) exhausts the budget.
x2*
x1* x1

7. Optimal Choice (with two goods case)- Graphical Method
Optimal Condition : Both conditions below should hold
Tangency Condition: If the demanded bundle (or optimal bundle) is an interior solution and indifference curves are in convex decreasing shape, then indifference curve should be tangent to the budget line at the optimal bundle ( i.e. slope of the budget line and slope of the indifference curve should be equal at the optimal bundle)
Budget Constraint: The budget is exhausted;

8. Optimal Choice for Cobb-Douglas Preferences
Suppose that the consumer has Cobb-Douglas preferences.

9. Optimal Choice for Cobb-Douglas Preferences
Then,
and
So the MRS is

10. Optimal Choice for Cobb-Douglas Preferences
Tangency Condition: Since MRS is diminishing and consumer likes both goods, IC’s are in convex decreasing shape. Hence, tangency condition should hold.

11. Optimal Choice for Cobb-Douglas Preferences
Budget Constraint: (x1*,x2*) also exhausts the budget so

12. Optimal Choice for Cobb-Douglas Preferences
Both tangency condition and budget constraint should be satisfied at the optimal.

13. Optimal Choice for Cobb-Douglas Preferences
So now we know that
(A)
Substitute
(B)
and get
This simplifies to ….

14. Optimal Choice for Cobb-Douglas Preferences
Substituting for x1* in
then gives

15. Optimal Choice for Cobb-Douglas Preferences
So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences is

16. Utility Maximization Problem and Optimal Choice (with two goods case) – Mathematical Method
As discussed in Math-review slides, we can use Lagrange Multiplier Method to solve Constraint Optimization problem.

17. Utility Maximization Problem and Optimal Choice (with two goods case) – Mathematical Method
From (1) and (2),
Also (3) should hold:
(4) is the same as tangency condition and (3) is as budget constraint in our Graphical Method.
Lagrange Method is more useful in n-Good case.

18. Optimal Choice Problem
We learned that as long as optimal choice is an interior solution and the indifference curves are convex decreasing, then the optimal can be determined by using tangency condition and budget constraint (and also by Lagrange Method)
BUT, some preferences (such as perfect complements, perfect substitutes, preferences with increasing MRS, etc.) do not have a convex decreasing Indifference Curves. So, tangency condition can not be used and also Lagrange method will not help us to find the solution.

19. Optimal Choice Problem when Indifference Curves are not smooth
Perfect Substitute: x2
CASE 1 :
IC1
IC2
Slope of BL = -p1/p2
Budget Line
Slope of IC= - MRS= - A/B x1

20. Optimal Choice for Perfect Substitute Preferencex2
CASE 2 :
IC1
IC2
Slope of IC= - MRS= - A/B
Budget Line
Slope of BL = -p1/p2 x1

21. Optimal Choice for Perfect Substitute Preferencex2
CASE 3 :
IC1
IC2
Slope of IC= - MRS= - A/B
Budget Line
Slope of BL = -p1/p2 x1

22. Optimal Choice for Perfect Complements Preferencex2
IC1
IC2
Optimal Bundle should satisfy both conditions below: x1

23. Optimal Choice for Perfect Complements Preference

24. Optimal Choice for Preferences with increasing MRSx2
Which is the most preferred affordable bundle? x1

25. Optimal Choice for Preferences with increasing MRSx2
The most preferred affordable bundle x1

26. Optimal Choice for Preferences with increasing MRS
Notice that the “tangency solution” is not the most preferred affordable bundle. x2
The most preferred affordable bundle x1